// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008-2011 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2009 Keir Mierle <mierle@gmail.com>
// Copyright (C) 2009 Benoit Jacob <jacob.benoit.1@gmail.com>
// Copyright (C) 2011 Timothy E. Holy <tim.holy@gmail.com >
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_LDLT_H
#define EIGEN_LDLT_H

namespace Eigen {

namespace internal {
    template <typename _MatrixType, int _UpLo> struct traits<LDLT<_MatrixType, _UpLo>> : traits<_MatrixType>
    {
        typedef MatrixXpr XprKind;
        typedef SolverStorage StorageKind;
        typedef int StorageIndex;
        enum
        {
            Flags = 0
        };
    };

    template <typename MatrixType, int UpLo> struct LDLT_Traits;

    // PositiveSemiDef means positive semi-definite and non-zero; same for NegativeSemiDef
    enum SignMatrix
    {
        PositiveSemiDef,
        NegativeSemiDef,
        ZeroSign,
        Indefinite
    };
}  // namespace internal

/** \ingroup Cholesky_Module
  *
  * \class LDLT
  *
  * \brief Robust Cholesky decomposition of a matrix with pivoting
  *
  * \tparam _MatrixType the type of the matrix of which to compute the LDL^T Cholesky decomposition
  * \tparam _UpLo the triangular part that will be used for the decompositon: Lower (default) or Upper.
  *             The other triangular part won't be read.
  *
  * Perform a robust Cholesky decomposition of a positive semidefinite or negative semidefinite
  * matrix \f$ A \f$ such that \f$ A =  P^TLDL^*P \f$, where P is a permutation matrix, L
  * is lower triangular with a unit diagonal and D is a diagonal matrix.
  *
  * The decomposition uses pivoting to ensure stability, so that D will have
  * zeros in the bottom right rank(A) - n submatrix. Avoiding the square root
  * on D also stabilizes the computation.
  *
  * Remember that Cholesky decompositions are not rank-revealing. Also, do not use a Cholesky
  * decomposition to determine whether a system of equations has a solution.
  *
  * This class supports the \link InplaceDecomposition inplace decomposition \endlink mechanism.
  *
  * \sa MatrixBase::ldlt(), SelfAdjointView::ldlt(), class LLT
  */
template <typename _MatrixType, int _UpLo> class LDLT : public SolverBase<LDLT<_MatrixType, _UpLo>>
{
public:
    typedef _MatrixType MatrixType;
    typedef SolverBase<LDLT> Base;
    friend class SolverBase<LDLT>;

    EIGEN_GENERIC_PUBLIC_INTERFACE(LDLT)
    enum
    {
        MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
        MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
        UpLo = _UpLo
    };
    typedef Matrix<Scalar, RowsAtCompileTime, 1, 0, MaxRowsAtCompileTime, 1> TmpMatrixType;

    typedef Transpositions<RowsAtCompileTime, MaxRowsAtCompileTime> TranspositionType;
    typedef PermutationMatrix<RowsAtCompileTime, MaxRowsAtCompileTime> PermutationType;

    typedef internal::LDLT_Traits<MatrixType, UpLo> Traits;

    /** \brief Default Constructor.
      *
      * The default constructor is useful in cases in which the user intends to
      * perform decompositions via LDLT::compute(const MatrixType&).
      */
    LDLT() : m_matrix(), m_transpositions(), m_sign(internal::ZeroSign), m_isInitialized(false) {}

    /** \brief Default Constructor with memory preallocation
      *
      * Like the default constructor but with preallocation of the internal data
      * according to the specified problem \a size.
      * \sa LDLT()
      */
    explicit LDLT(Index size) : m_matrix(size, size), m_transpositions(size), m_temporary(size), m_sign(internal::ZeroSign), m_isInitialized(false) {}

    /** \brief Constructor with decomposition
      *
      * This calculates the decomposition for the input \a matrix.
      *
      * \sa LDLT(Index size)
      */
    template <typename InputType>
    explicit LDLT(const EigenBase<InputType>& matrix)
        : m_matrix(matrix.rows(), matrix.cols()), m_transpositions(matrix.rows()), m_temporary(matrix.rows()), m_sign(internal::ZeroSign),
          m_isInitialized(false)
    {
        compute(matrix.derived());
    }

    /** \brief Constructs a LDLT factorization from a given matrix
      *
      * This overloaded constructor is provided for \link InplaceDecomposition inplace decomposition \endlink when \c MatrixType is a Eigen::Ref.
      *
      * \sa LDLT(const EigenBase&)
      */
    template <typename InputType>
    explicit LDLT(EigenBase<InputType>& matrix)
        : m_matrix(matrix.derived()), m_transpositions(matrix.rows()), m_temporary(matrix.rows()), m_sign(internal::ZeroSign), m_isInitialized(false)
    {
        compute(matrix.derived());
    }

    /** Clear any existing decomposition
     * \sa rankUpdate(w,sigma)
     */
    void setZero() { m_isInitialized = false; }

    /** \returns a view of the upper triangular matrix U */
    inline typename Traits::MatrixU matrixU() const
    {
        eigen_assert(m_isInitialized && "LDLT is not initialized.");
        return Traits::getU(m_matrix);
    }

    /** \returns a view of the lower triangular matrix L */
    inline typename Traits::MatrixL matrixL() const
    {
        eigen_assert(m_isInitialized && "LDLT is not initialized.");
        return Traits::getL(m_matrix);
    }

    /** \returns the permutation matrix P as a transposition sequence.
      */
    inline const TranspositionType& transpositionsP() const
    {
        eigen_assert(m_isInitialized && "LDLT is not initialized.");
        return m_transpositions;
    }

    /** \returns the coefficients of the diagonal matrix D */
    inline Diagonal<const MatrixType> vectorD() const
    {
        eigen_assert(m_isInitialized && "LDLT is not initialized.");
        return m_matrix.diagonal();
    }

    /** \returns true if the matrix is positive (semidefinite) */
    inline bool isPositive() const
    {
        eigen_assert(m_isInitialized && "LDLT is not initialized.");
        return m_sign == internal::PositiveSemiDef || m_sign == internal::ZeroSign;
    }

    /** \returns true if the matrix is negative (semidefinite) */
    inline bool isNegative(void) const
    {
        eigen_assert(m_isInitialized && "LDLT is not initialized.");
        return m_sign == internal::NegativeSemiDef || m_sign == internal::ZeroSign;
    }

#ifdef EIGEN_PARSED_BY_DOXYGEN
    /** \returns a solution x of \f$ A x = b \f$ using the current decomposition of A.
      *
      * This function also supports in-place solves using the syntax <tt>x = decompositionObject.solve(x)</tt> .
      *
      * \note_about_checking_solutions
      *
      * More precisely, this method solves \f$ A x = b \f$ using the decomposition \f$ A = P^T L D L^* P \f$
      * by solving the systems \f$ P^T y_1 = b \f$, \f$ L y_2 = y_1 \f$, \f$ D y_3 = y_2 \f$,
      * \f$ L^* y_4 = y_3 \f$ and \f$ P x = y_4 \f$ in succession. If the matrix \f$ A \f$ is singular, then
      * \f$ D \f$ will also be singular (all the other matrices are invertible). In that case, the
      * least-square solution of \f$ D y_3 = y_2 \f$ is computed. This does not mean that this function
      * computes the least-square solution of \f$ A x = b \f$ if \f$ A \f$ is singular.
      *
      * \sa MatrixBase::ldlt(), SelfAdjointView::ldlt()
      */
    template <typename Rhs> inline const Solve<LDLT, Rhs> solve(const MatrixBase<Rhs>& b) const;
#endif

    template <typename Derived> bool solveInPlace(MatrixBase<Derived>& bAndX) const;

    template <typename InputType> LDLT& compute(const EigenBase<InputType>& matrix);

    /** \returns an estimate of the reciprocal condition number of the matrix of
     *  which \c *this is the LDLT decomposition.
     */
    RealScalar rcond() const
    {
        eigen_assert(m_isInitialized && "LDLT is not initialized.");
        return internal::rcond_estimate_helper(m_l1_norm, *this);
    }

    template <typename Derived> LDLT& rankUpdate(const MatrixBase<Derived>& w, const RealScalar& alpha = 1);

    /** \returns the internal LDLT decomposition matrix
      *
      * TODO: document the storage layout
      */
    inline const MatrixType& matrixLDLT() const
    {
        eigen_assert(m_isInitialized && "LDLT is not initialized.");
        return m_matrix;
    }

    MatrixType reconstructedMatrix() const;

    /** \returns the adjoint of \c *this, that is, a const reference to the decomposition itself as the underlying matrix is self-adjoint.
      *
      * This method is provided for compatibility with other matrix decompositions, thus enabling generic code such as:
      * \code x = decomposition.adjoint().solve(b) \endcode
      */
    const LDLT& adjoint() const { return *this; };

    EIGEN_DEVICE_FUNC inline EIGEN_CONSTEXPR Index rows() const EIGEN_NOEXCEPT { return m_matrix.rows(); }
    EIGEN_DEVICE_FUNC inline EIGEN_CONSTEXPR Index cols() const EIGEN_NOEXCEPT { return m_matrix.cols(); }

    /** \brief Reports whether previous computation was successful.
      *
      * \returns \c Success if computation was successful,
      *          \c NumericalIssue if the factorization failed because of a zero pivot.
      */
    ComputationInfo info() const
    {
        eigen_assert(m_isInitialized && "LDLT is not initialized.");
        return m_info;
    }

#ifndef EIGEN_PARSED_BY_DOXYGEN
    template <typename RhsType, typename DstType> void _solve_impl(const RhsType& rhs, DstType& dst) const;

    template <bool Conjugate, typename RhsType, typename DstType> void _solve_impl_transposed(const RhsType& rhs, DstType& dst) const;
#endif

protected:
    static void check_template_parameters() { EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar); }

    /** \internal
      * Used to compute and store the Cholesky decomposition A = L D L^* = U^* D U.
      * The strict upper part is used during the decomposition, the strict lower
      * part correspond to the coefficients of L (its diagonal is equal to 1 and
      * is not stored), and the diagonal entries correspond to D.
      */
    MatrixType m_matrix;
    RealScalar m_l1_norm;
    TranspositionType m_transpositions;
    TmpMatrixType m_temporary;
    internal::SignMatrix m_sign;
    bool m_isInitialized;
    ComputationInfo m_info;
};

namespace internal {

    template <int UpLo> struct ldlt_inplace;

    template <> struct ldlt_inplace<Lower>
    {
        template <typename MatrixType, typename TranspositionType, typename Workspace>
        static bool unblocked(MatrixType& mat, TranspositionType& transpositions, Workspace& temp, SignMatrix& sign)
        {
            using std::abs;
            typedef typename MatrixType::Scalar Scalar;
            typedef typename MatrixType::RealScalar RealScalar;
            typedef typename TranspositionType::StorageIndex IndexType;
            eigen_assert(mat.rows() == mat.cols());
            const Index size = mat.rows();
            bool found_zero_pivot = false;
            bool ret = true;

            if (size <= 1)
            {
                transpositions.setIdentity();
                if (size == 0)
                    sign = ZeroSign;
                else if (numext::real(mat.coeff(0, 0)) > static_cast<RealScalar>(0))
                    sign = PositiveSemiDef;
                else if (numext::real(mat.coeff(0, 0)) < static_cast<RealScalar>(0))
                    sign = NegativeSemiDef;
                else
                    sign = ZeroSign;
                return true;
            }

            for (Index k = 0; k < size; ++k)
            {
                // Find largest diagonal element
                Index index_of_biggest_in_corner;
                mat.diagonal().tail(size - k).cwiseAbs().maxCoeff(&index_of_biggest_in_corner);
                index_of_biggest_in_corner += k;

                transpositions.coeffRef(k) = IndexType(index_of_biggest_in_corner);
                if (k != index_of_biggest_in_corner)
                {
                    // apply the transposition while taking care to consider only
                    // the lower triangular part
                    Index s = size - index_of_biggest_in_corner - 1;  // trailing size after the biggest element
                    mat.row(k).head(k).swap(mat.row(index_of_biggest_in_corner).head(k));
                    mat.col(k).tail(s).swap(mat.col(index_of_biggest_in_corner).tail(s));
                    std::swap(mat.coeffRef(k, k), mat.coeffRef(index_of_biggest_in_corner, index_of_biggest_in_corner));
                    for (Index i = k + 1; i < index_of_biggest_in_corner; ++i)
                    {
                        Scalar tmp = mat.coeffRef(i, k);
                        mat.coeffRef(i, k) = numext::conj(mat.coeffRef(index_of_biggest_in_corner, i));
                        mat.coeffRef(index_of_biggest_in_corner, i) = numext::conj(tmp);
                    }
                    if (NumTraits<Scalar>::IsComplex)
                        mat.coeffRef(index_of_biggest_in_corner, k) = numext::conj(mat.coeff(index_of_biggest_in_corner, k));
                }

                // partition the matrix:
                //       A00 |  -  |  -
                // lu  = A10 | A11 |  -
                //       A20 | A21 | A22
                Index rs = size - k - 1;
                Block<MatrixType, Dynamic, 1> A21(mat, k + 1, k, rs, 1);
                Block<MatrixType, 1, Dynamic> A10(mat, k, 0, 1, k);
                Block<MatrixType, Dynamic, Dynamic> A20(mat, k + 1, 0, rs, k);

                if (k > 0)
                {
                    temp.head(k) = mat.diagonal().real().head(k).asDiagonal() * A10.adjoint();
                    mat.coeffRef(k, k) -= (A10 * temp.head(k)).value();
                    if (rs > 0)
                        A21.noalias() -= A20 * temp.head(k);
                }

                // In some previous versions of Eigen (e.g., 3.2.1), the scaling was omitted if the pivot
                // was smaller than the cutoff value. However, since LDLT is not rank-revealing
                // we should only make sure that we do not introduce INF or NaN values.
                // Remark that LAPACK also uses 0 as the cutoff value.
                RealScalar realAkk = numext::real(mat.coeffRef(k, k));
                bool pivot_is_valid = (abs(realAkk) > RealScalar(0));

                if (k == 0 && !pivot_is_valid)
                {
                    // The entire diagonal is zero, there is nothing more to do
                    // except filling the transpositions, and checking whether the matrix is zero.
                    sign = ZeroSign;
                    for (Index j = 0; j < size; ++j)
                    {
                        transpositions.coeffRef(j) = IndexType(j);
                        ret = ret && (mat.col(j).tail(size - j - 1).array() == Scalar(0)).all();
                    }
                    return ret;
                }

                if ((rs > 0) && pivot_is_valid)
                    A21 /= realAkk;
                else if (rs > 0)
                    ret = ret && (A21.array() == Scalar(0)).all();

                if (found_zero_pivot && pivot_is_valid)
                    ret = false;  // factorization failed
                else if (!pivot_is_valid)
                    found_zero_pivot = true;

                if (sign == PositiveSemiDef)
                {
                    if (realAkk < static_cast<RealScalar>(0))
                        sign = Indefinite;
                }
                else if (sign == NegativeSemiDef)
                {
                    if (realAkk > static_cast<RealScalar>(0))
                        sign = Indefinite;
                }
                else if (sign == ZeroSign)
                {
                    if (realAkk > static_cast<RealScalar>(0))
                        sign = PositiveSemiDef;
                    else if (realAkk < static_cast<RealScalar>(0))
                        sign = NegativeSemiDef;
                }
            }

            return ret;
        }

        // Reference for the algorithm: Davis and Hager, "Multiple Rank
        // Modifications of a Sparse Cholesky Factorization" (Algorithm 1)
        // Trivial rearrangements of their computations (Timothy E. Holy)
        // allow their algorithm to work for rank-1 updates even if the
        // original matrix is not of full rank.
        // Here only rank-1 updates are implemented, to reduce the
        // requirement for intermediate storage and improve accuracy
        template <typename MatrixType, typename WDerived>
        static bool updateInPlace(MatrixType& mat, MatrixBase<WDerived>& w, const typename MatrixType::RealScalar& sigma = 1)
        {
            using numext::isfinite;
            typedef typename MatrixType::Scalar Scalar;
            typedef typename MatrixType::RealScalar RealScalar;

            const Index size = mat.rows();
            eigen_assert(mat.cols() == size && w.size() == size);

            RealScalar alpha = 1;

            // Apply the update
            for (Index j = 0; j < size; j++)
            {
                // Check for termination due to an original decomposition of low-rank
                if (!(isfinite)(alpha))
                    break;

                // Update the diagonal terms
                RealScalar dj = numext::real(mat.coeff(j, j));
                Scalar wj = w.coeff(j);
                RealScalar swj2 = sigma * numext::abs2(wj);
                RealScalar gamma = dj * alpha + swj2;

                mat.coeffRef(j, j) += swj2 / alpha;
                alpha += swj2 / dj;

                // Update the terms of L
                Index rs = size - j - 1;
                w.tail(rs) -= wj * mat.col(j).tail(rs);
                if (gamma != 0)
                    mat.col(j).tail(rs) += (sigma * numext::conj(wj) / gamma) * w.tail(rs);
            }
            return true;
        }

        template <typename MatrixType, typename TranspositionType, typename Workspace, typename WType>
        static bool
        update(MatrixType& mat, const TranspositionType& transpositions, Workspace& tmp, const WType& w, const typename MatrixType::RealScalar& sigma = 1)
        {
            // Apply the permutation to the input w
            tmp = transpositions * w;

            return ldlt_inplace<Lower>::updateInPlace(mat, tmp, sigma);
        }
    };

    template <> struct ldlt_inplace<Upper>
    {
        template <typename MatrixType, typename TranspositionType, typename Workspace>
        static EIGEN_STRONG_INLINE bool unblocked(MatrixType& mat, TranspositionType& transpositions, Workspace& temp, SignMatrix& sign)
        {
            Transpose<MatrixType> matt(mat);
            return ldlt_inplace<Lower>::unblocked(matt, transpositions, temp, sign);
        }

        template <typename MatrixType, typename TranspositionType, typename Workspace, typename WType>
        static EIGEN_STRONG_INLINE bool
        update(MatrixType& mat, TranspositionType& transpositions, Workspace& tmp, WType& w, const typename MatrixType::RealScalar& sigma = 1)
        {
            Transpose<MatrixType> matt(mat);
            return ldlt_inplace<Lower>::update(matt, transpositions, tmp, w.conjugate(), sigma);
        }
    };

    template <typename MatrixType> struct LDLT_Traits<MatrixType, Lower>
    {
        typedef const TriangularView<const MatrixType, UnitLower> MatrixL;
        typedef const TriangularView<const typename MatrixType::AdjointReturnType, UnitUpper> MatrixU;
        static inline MatrixL getL(const MatrixType& m) { return MatrixL(m); }
        static inline MatrixU getU(const MatrixType& m) { return MatrixU(m.adjoint()); }
    };

    template <typename MatrixType> struct LDLT_Traits<MatrixType, Upper>
    {
        typedef const TriangularView<const typename MatrixType::AdjointReturnType, UnitLower> MatrixL;
        typedef const TriangularView<const MatrixType, UnitUpper> MatrixU;
        static inline MatrixL getL(const MatrixType& m) { return MatrixL(m.adjoint()); }
        static inline MatrixU getU(const MatrixType& m) { return MatrixU(m); }
    };

}  // end namespace internal

/** Compute / recompute the LDLT decomposition A = L D L^* = U^* D U of \a matrix
  */
template <typename MatrixType, int _UpLo> template <typename InputType> LDLT<MatrixType, _UpLo>& LDLT<MatrixType, _UpLo>::compute(const EigenBase<InputType>& a)
{
    check_template_parameters();

    eigen_assert(a.rows() == a.cols());
    const Index size = a.rows();

    m_matrix = a.derived();

    // Compute matrix L1 norm = max abs column sum.
    m_l1_norm = RealScalar(0);
    // TODO move this code to SelfAdjointView
    for (Index col = 0; col < size; ++col)
    {
        RealScalar abs_col_sum;
        if (_UpLo == Lower)
            abs_col_sum = m_matrix.col(col).tail(size - col).template lpNorm<1>() + m_matrix.row(col).head(col).template lpNorm<1>();
        else
            abs_col_sum = m_matrix.col(col).head(col).template lpNorm<1>() + m_matrix.row(col).tail(size - col).template lpNorm<1>();
        if (abs_col_sum > m_l1_norm)
            m_l1_norm = abs_col_sum;
    }

    m_transpositions.resize(size);
    m_isInitialized = false;
    m_temporary.resize(size);
    m_sign = internal::ZeroSign;

    m_info = internal::ldlt_inplace<UpLo>::unblocked(m_matrix, m_transpositions, m_temporary, m_sign) ? Success : NumericalIssue;

    m_isInitialized = true;
    return *this;
}

/** Update the LDLT decomposition:  given A = L D L^T, efficiently compute the decomposition of A + sigma w w^T.
 * \param w a vector to be incorporated into the decomposition.
 * \param sigma a scalar, +1 for updates and -1 for "downdates," which correspond to removing previously-added column vectors. Optional; default value is +1.
 * \sa setZero()
  */
template <typename MatrixType, int _UpLo>
template <typename Derived>
LDLT<MatrixType, _UpLo>& LDLT<MatrixType, _UpLo>::rankUpdate(const MatrixBase<Derived>& w, const typename LDLT<MatrixType, _UpLo>::RealScalar& sigma)
{
    typedef typename TranspositionType::StorageIndex IndexType;
    const Index size = w.rows();
    if (m_isInitialized)
    {
        eigen_assert(m_matrix.rows() == size);
    }
    else
    {
        m_matrix.resize(size, size);
        m_matrix.setZero();
        m_transpositions.resize(size);
        for (Index i = 0; i < size; i++) m_transpositions.coeffRef(i) = IndexType(i);
        m_temporary.resize(size);
        m_sign = sigma >= 0 ? internal::PositiveSemiDef : internal::NegativeSemiDef;
        m_isInitialized = true;
    }

    internal::ldlt_inplace<UpLo>::update(m_matrix, m_transpositions, m_temporary, w, sigma);

    return *this;
}

#ifndef EIGEN_PARSED_BY_DOXYGEN
template <typename _MatrixType, int _UpLo>
template <typename RhsType, typename DstType>
void LDLT<_MatrixType, _UpLo>::_solve_impl(const RhsType& rhs, DstType& dst) const
{
    _solve_impl_transposed<true>(rhs, dst);
}

template <typename _MatrixType, int _UpLo>
template <bool Conjugate, typename RhsType, typename DstType>
void LDLT<_MatrixType, _UpLo>::_solve_impl_transposed(const RhsType& rhs, DstType& dst) const
{
    // dst = P b
    dst = m_transpositions * rhs;

    // dst = L^-1 (P b)
    // dst = L^-*T (P b)
    matrixL().template conjugateIf<!Conjugate>().solveInPlace(dst);

    // dst = D^-* (L^-1 P b)
    // dst = D^-1 (L^-*T P b)
    // more precisely, use pseudo-inverse of D (see bug 241)
    using std::abs;
    const typename Diagonal<const MatrixType>::RealReturnType vecD(vectorD());
    // In some previous versions, tolerance was set to the max of 1/highest (or rather numeric_limits::min())
    // and the maximal diagonal entry * epsilon as motivated by LAPACK's xGELSS:
    // RealScalar tolerance = numext::maxi(vecD.array().abs().maxCoeff() * NumTraits<RealScalar>::epsilon(),RealScalar(1) / NumTraits<RealScalar>::highest());
    // However, LDLT is not rank revealing, and so adjusting the tolerance wrt to the highest
    // diagonal element is not well justified and leads to numerical issues in some cases.
    // Moreover, Lapack's xSYTRS routines use 0 for the tolerance.
    // Using numeric_limits::min() gives us more robustness to denormals.
    RealScalar tolerance = (std::numeric_limits<RealScalar>::min)();
    for (Index i = 0; i < vecD.size(); ++i)
    {
        if (abs(vecD(i)) > tolerance)
            dst.row(i) /= vecD(i);
        else
            dst.row(i).setZero();
    }

    // dst = L^-* (D^-* L^-1 P b)
    // dst = L^-T (D^-1 L^-*T P b)
    matrixL().transpose().template conjugateIf<Conjugate>().solveInPlace(dst);

    // dst = P^T (L^-* D^-* L^-1 P b) = A^-1 b
    // dst = P^-T (L^-T D^-1 L^-*T P b) = A^-1 b
    dst = m_transpositions.transpose() * dst;
}
#endif

/** \internal use x = ldlt_object.solve(x);
  *
  * This is the \em in-place version of solve().
  *
  * \param bAndX represents both the right-hand side matrix b and result x.
  *
  * \returns true always! If you need to check for existence of solutions, use another decomposition like LU, QR, or SVD.
  *
  * This version avoids a copy when the right hand side matrix b is not
  * needed anymore.
  *
  * \sa LDLT::solve(), MatrixBase::ldlt()
  */
template <typename MatrixType, int _UpLo> template <typename Derived> bool LDLT<MatrixType, _UpLo>::solveInPlace(MatrixBase<Derived>& bAndX) const
{
    eigen_assert(m_isInitialized && "LDLT is not initialized.");
    eigen_assert(m_matrix.rows() == bAndX.rows());

    bAndX = this->solve(bAndX);

    return true;
}

/** \returns the matrix represented by the decomposition,
 * i.e., it returns the product: P^T L D L^* P.
 * This function is provided for debug purpose. */
template <typename MatrixType, int _UpLo> MatrixType LDLT<MatrixType, _UpLo>::reconstructedMatrix() const
{
    eigen_assert(m_isInitialized && "LDLT is not initialized.");
    const Index size = m_matrix.rows();
    MatrixType res(size, size);

    // P
    res.setIdentity();
    res = transpositionsP() * res;
    // L^* P
    res = matrixU() * res;
    // D(L^*P)
    res = vectorD().real().asDiagonal() * res;
    // L(DL^*P)
    res = matrixL() * res;
    // P^T (LDL^*P)
    res = transpositionsP().transpose() * res;

    return res;
}

/** \cholesky_module
  * \returns the Cholesky decomposition with full pivoting without square root of \c *this
  * \sa MatrixBase::ldlt()
  */
template <typename MatrixType, unsigned int UpLo>
inline const LDLT<typename SelfAdjointView<MatrixType, UpLo>::PlainObject, UpLo> SelfAdjointView<MatrixType, UpLo>::ldlt() const
{
    return LDLT<PlainObject, UpLo>(m_matrix);
}

/** \cholesky_module
  * \returns the Cholesky decomposition with full pivoting without square root of \c *this
  * \sa SelfAdjointView::ldlt()
  */
template <typename Derived> inline const LDLT<typename MatrixBase<Derived>::PlainObject> MatrixBase<Derived>::ldlt() const
{
    return LDLT<PlainObject>(derived());
}

}  // end namespace Eigen

#endif  // EIGEN_LDLT_H
